1917] AT THE SURFACE OF DEEP WATER 481 When we introduce the values o sufficient approximation, we have 9^ 769a5) 8 192 } and 7, already determined in (7) with cos 2# -| -JT + 2i lla 303 315a5| __ tf 8 + 128 J 3"' 125a5 c /11V cos 5*,... (11) 6 8 . 128 3 ------ 384 in agreement with equations (13), (18) of my former paper when allowance is made for the different suppositions with respect to ^} as may be effected by expressing both results in terms of a, the coefficient of cos x, instead of a. The next step is the further development of the pressure equation (2), so as to include terms of the order a7. Where /9, 7, etc. occur as factors, the expression for y to the third order, as in (5), suffices; but a more accurate value is required in ct2e~22/. Expanding the exponentials and replacing products of cosines by cosines of sums and differences, we find in the first place A B + cos x \ - 4aB -f- + cos3aj-^-2Ğ/3 + 47-?^+i^ 1 O TQ f Trf cos 6Ğ7 + 68 > 25a3/3 15a27 From the terms in cos a? we now eliminate cos x by means of a cos a? = y (1 - fa2) + £a2 + |a2 cos 20 f, thus altering those terras of (12) which are constant, and which contain y and cos 2#. Thus modified, (12) becomes f 4aB J -jr [* The terms in o3/3(cosa;, cos3a;) should read +-g-a3/8cosa;) +1 a3^ cos 3s; apparently the term - 4a8j3 cos a; cos Zx had been omitted from the development of 2^e-2^ cos 2x. il t Since terms of order a7 are retained, the term -= a3 cos Bx should be added to the expression for a cos a;., W. P. S.] R. VI. 31......................................... (10) 2gy = 1 + 2 (1 - g) y + cPe'*" + 2/3e~^ cos 2a?